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Coarse-graining Hamiltonian systems using WSINDy.

Daniel A Messenger1, Joshua W Burby2, David M Bortz3

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Summary
This summary is machine-generated.

Weak form equation learning, a method called WSINDy, efficiently identifies reduced Hamiltonian systems. This approach is robust to noise and perturbations, making it ideal for coarse-graining complex dynamics.

Keywords:
Coarse-grainingHamiltonian systemsWSINDy

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Area of Science:

  • Computational Physics
  • Dynamical Systems
  • Applied Mathematics

Background:

  • Weak form equation learning and surrogate modeling are computationally efficient and robust for discovering dynamics governed by ODEs, PDEs, and SDEs.
  • Coarse-graining techniques like homogenization and mean-field descriptions are used for interacting particle systems.
  • Hamiltonian dynamics with approximate symmetries, often linked to timescale separation, present challenges in deriving reduced-order models.

Purpose of the Study:

  • To extend weak form equation learning for coarse-graining Hamiltonian dynamics with approximate symmetries.
  • To demonstrate the capability of WSINDy (Weak-form Sparse Identification of Nonlinear Dynamics) in identifying reduced Hamiltonian systems.
  • To provide theoretical justification for the method's effectiveness in Hamiltonian coarse-graining.

Main Methods:

  • Utilized WSINDy to identify reduced Hamiltonian systems from data, leveraging its ability to preserve Hamiltonian structure by restricting to a basis of Hamiltonian vector fields.
  • Employed a single trajectory for learning the global reduced Hamiltonian, avoiding computationally expensive forward solves.
  • Applied the method to nearly-periodic Hamiltonian systems exhibiting approximate symmetries.

Main Results:

  • WSINDy successfully identified reduced Hamiltonian systems, even with large perturbations and extrinsic noise.
  • The method achieved dimension reduction by at least two, accurately capturing the leading-order dynamics.
  • A theoretical contribution proved that first-order averaging preserves Hamiltonian structure in nearly-periodic Hamiltonian systems, justifying the WSINDy approach.

Conclusions:

  • Weak form equation learning, specifically WSINDy, is a computationally efficient and robust method for Hamiltonian coarse-graining.
  • The approach effectively identifies reduced-order models for systems with approximate symmetries, preserving the underlying Hamiltonian structure.
  • The method's efficacy was illustrated through physically relevant examples, including coupled oscillators and charged particle dynamics.